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YOU are better than YOU think. Show yourself how:
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-/[]\- Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite. Logic
Mastery Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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-/[]\- What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts. Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice. |
Four Skeptical Essays on Constructivism
1. Constructivist Hopes and Inspiration for Education, Wonderful and Exhilarating, but the Manual is Overdue
The instructor job besides showing students current rules and practices to try and follow is to check and correct their mastery of the latter in student work. The arrangement and uses of rules and practices in mathematics, developed and discovered over time by trial and error through paths not all optimal, need to be presented to students (a) directly or (b) through clues that make current rules and practices simple to see and grasp. Site material helps with direct instruction, option (a). The constructivist and learning-by-discovery movements in education, advocate the indirect instruction, option (b) strongly as the best route for learning, but do not fully say how. Option (b) has the potential to engage and motivate students, to develop skills and confidence, and as instructors we should look for examples of option (b) to use our instruction. The constructivist and learning-by-discovery advocates in education having developed their theory and hopes for its success now need to pay attentions to details and provide a manual. Since 1990 and the advent of the Canadian & United States NCTM constructivist manifesto in mathematics, its 1989 standards, the manual is overdue. While more and more examples are being developed and shared, the manual today (2007) is still overdue and incomplete. Whence mathematics instruction in North America is driven by hope and inspiration, but not detail. 2. Constructivism versus Direct InstructionSite material is written by a former college level instructor in the habit of trying to be the sage on stage (easier said than done in the past, but now more easily done) telling students what is expected by providing clear objectives, and then trying to develop those skills directly and clearly. Today, there is another fashion in education, especially at the primary and secondary level in which instructors are asked to find activities, situations and applications, which lead students to discover and become interested in mathematics, and so meet the objectives that used to be or could have stated and developed directly if not always clearly. I would like to see the new fashion combined with a clear end of course summary and consolidation of the objectives, so that student see how those objectives could have reach directly. Then students would enjoy the best of direct and indirect instruction. The prerequisite and safety-net for the new fashion is of course the ability to explain or develop all skills and concepts directly and clearly, with mastery of none left to chance. A subject is not understood fully until students & teachers can be shown how to develop its skills and concepts inductively and deductively in others in a repeatable and reproducible, and therefore reproducible manner. The logical development or present and changing structure of modern scientific or technical discipline, mathematics included, resembles that of a sequence of constructions in which earlier construction serve as scaffolding for the current structure, scaffolding that may not remain as part of the current self-supporting edifice or construction, and so the history of the discipline is lost. Direct instruction in the past has had it flaws. The observation or conviction that mathematics mastery is a natural talent points to an old incompleteness in the explanation or exposition of skills and concepts. Site coverage or introduction of logic, algebra and even calculus fills in a few gaps in the exposition to provide a firmer base for learning and teaching, the lack of which has slowed or harmed course design. Most marks in mathematics come from written work. Errors in notation, logic and comprehension, incoherency and confusion, can be observed. Students need to learn the definitions and methods in mathematics so that they can combine the definitions and rules, one at a time and one after, carefully, precisely, logically and creatively to arrive at results that are repeatable and reproducible, and hence verifiable. Instructors have a duty to catch & gently correct errors in notation and comprehension indicated by written work, so student may learn from their mistakes, and learn how to recognize mistakes their reasoning and how to test their conclusions. While the mathematics teacher cannot read the mind of student to see inner workings, the student can demonstrate skills and knowledge through written work and becoming a tutor or teacher.
Cognitive or constructivist theories for education have many faces. The most appealing face calls for activities, the introduction of situations and applications, in which students see the need for skills and concepts, and want to learn them. That is an appealing alternative, if one can find it, to direct instruction. That positive aspect of constructivism could be combined with direct instruction. Activities involving situations or applications may engage students and give a context and driving force for meeting the objectives of direct instruction.
A second face, one I oppose, emphasizes and favors the subjective nature of knowledge, and say such knowledge is not for testing nor correction because testing is not reliable/. This face literally push aside or turns upside down the empirical method for discovery and verification of method-based knowledge in science, technology and business in which observable, repeatable and reproducible methods are sought. In education in my view is an empirical art. In it, students first learn via trial, error and reason to give teachers what the teacher or syllabus wants before or besides demonstrating their creative skills There is a common body of rules and patterns to be met and mastered, directly or not. A third face of constructivism lies in its advocacy - the existence of textbooks and advocates without domain expertise who prescribe for all domains the constructivist approach for application before methods for the latter have tried and tested in any. This dogmatism in its application is not appealing and a recipe for disaster. The call for research-based methods in education that advocates the widespread use of peer-reviewed conjectures in the classroom without testing, or regardless of the results of testing, is impractical. Just as drug companies are asked to do field testing, so should pushers of educational reform.
A fourth face of constructivism within mathematics lies in its advocacy of or collusion with the use of calculators and technology. While technology can facilitate calculations and remove the drudgery of calculation with large quantities of data, the mastery of mathematics at the college level and the applications of mathematics in society at large still require an intellectual mastery of operations with whole numbers and fractions. Advance mathematics, say calculus and beyond, requires a mastery of functions, trig, algebra, geometry, logic and numbers in primary and secondary schools. Creeping reforms which push aside or de-emphasize the efficient mastery of fraction (fraction sense and operation) and associated number theory (primes, lcd, gcd) dilute the arithmetic base on which algebra, trig and calculus rely. The ignorant de-emphasize of arithmetic skill development has a downstream dulling effect on high school mathematics education. Subject experts have no say.
3. More On Constructivism:
The logical connection and development of high school mathematics and calculus springs from the minds of many - key individual included. The foregoing defines paths for understanding, individually followed, but developed and refined over decades or centuries. To avoid re-inventing the wheel time and time again, students and teachers need expositions which define and develop skills and concepts clearly and directly. Problem solving in mathematics is enhanced by the identification and mastery of key skills and concepts, and the emphasis of a jigsaw puzzle approaches in which the easiest parts of a puzzle are identified and tackled first. While the ability to construct comprehension directly from hands-on activities is important, the ability to learn from second-hand experience, that summarized in hopefully direct and clear expositions or lessons, is also important. Education may involve a mix of student inquiry plus teacher direction or feedback, with the latter being authoritative as teacher provide evaluation and marks. Student need to learn how to do and reason in a repeatable and reproducible manner. Anything less points to an inconsistent mastery of know-how and know-why. And students need to be corrected. The notion that student notions and answers are valid because they are individually constructed represents a low standard for education. Learning by trial and error is an authentic, realistic and essential part of business, science, technology and decision-making. Where errors are not identified, learning will not occur.
June 3, 2006. 4. Constructivism Flaws - Difficulties with Discovery Method
While people should construct their own knowledge as much as possible, in empirical and/or mathematical arts and disciplines, students need to meet rules and patterns previously found, whose discovery was not obvious, and whose verification in the classroom may be partial. whose application needs to be practiced, so that skill and confidence follows in a repeatable and reproducible manner. Empirical arts and disciplines rely on nature to correct errors and identify the limits of current theories or explanations. In engineering and science, students are in the business of meeting methods and theories that can be used for design and prediction, or creating such theories. Every such method or theory is a gamble as failed predictions point to the need for an adaptation or rejection of the method or theory, while verified predictions may confirm (make more likely) but not prove the validity of an empirical theory. Mathematics knowledge appears to be empirical (a function of our senses) and appears to be logical (a function of ideas recorded and developed on paper). In mathematics, there two standards for correctness. First, in arithmetic, results should be repeatable and reproducible, and therefore verifiable. Second in the theoretical development and justification of methods and ideas, the latter need to be develop by direct and indirect chains of implication rules, staring from given axioms (assume patterns, gambles) in a repeatable, repeatable and therefore verifiable manner. The constructivist educational reform movement in calling for students to be engaged by authentic, genuine, authentic problems and situations echoes previous calls for reforms. Constructivist methods for engaging students via exploring and developing their prior knowledge of a subject are worth following. Yet constructivist rejection of tests or measurement of student abilities as being unreliable and being judgmental is empirically unsound. While students minds cannot be read, while measurement or observation of a skill or talent today is not guarantee that a student will maintain the same level of mastery tomorrow, and while tests may sample the student skills (a form of statistical quality control) instead of being comprehensive, education is an empirical art. The verification of student skills is a statistical affair. Reliability and feedback (correction) rises with the number of well-put observations. Training or skill development in empirical and logic based arts and disciplines aims for students mastery or display of skills and concepts in a repeatable and reproducible and hence verifiable fashion. Formation and evaluation in empirical arts and mathematics is or should be based on observation of student skills and practice. Performance standards need to be respected. The constructivist viewpoint that education should proceed by discovery, that knowledge is subjective, and hence not for correction, is inconsistent with the rule and pattern organization and development of mathematics and empirical arts and disciplines. The implication that personal knowledge and deeds are not for correction inconsistent with the judgmental, Euclidean rule and pattern based codification of geometry, modern mathematics and law. While constructivist methods are worth noting, the constructivist movement needs to be reconstructed in a rational fashion. Irrational parts, parts inconsistent with the hard sciences, need to be excised. |
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